The aim of this paper was to show that the Lagrange-d'Alembert and its equivalent the Gauss and Appel principle are not the only way to deduce the equations of motion of the nonholonomic systems. Instead of them we consider the generalization of the Hamiltonian principle for nonholonomic systems with non-zero transpositional relations. We apply this variational principle, which takes into the account transpositional relations different from the classical ones, and we deduce the equations of motion for the nonholonomic systems with constraints that in general are nonlinear in the velocity. These equations of motion coincide, except perhaps in a zero Lebesgue measure set, with the classical differential equations deduced with the d'Alembert-Lagrange principle. We provide a new point of view on the transpositional relations for the constrained mechanical systems: the virtual variations can produce zero or non-zero transpositional relations. In particular, the independent virtual variations can produce non-zero transpositional relations. For the unconstrained mechanical systems, the virtual variations always produce zero transpositional relations. We conjecture that the existence of the nonlinear constraints in the velocity must be sought outside of the Newtonian mechanics. We illustrate our results with examples.

Billiard trajectories inside an ellipsoid of Rn are tangent to n-1 quadrics of the pencil of confocal quadrics determined by the ellipsoid. The quadrics associated with periodic trajectories verify certain algebraic conditions. Cayley found them for the planar case. Dragovic and Radnovic generalized them to any dimension. We rewrite the original matrix formulation of these generalized Cayley conditions as a simpler polynomial one. We find several algebraic relations between caustic parameters and ellipsoidal parameters that give rise to non-singular periodic trajectories. These relations become remarkably simple when the elliptic period is minimal. We study the caustic types, the winding numbers and the ellipsoids of such minimal periodic trajectories. We also describe some non-minimal periodic trajectories

We give an upper bound for the maximum number N of algebraic limit cycles that a planar polynomial vector field of degree n can exhibit if the vector field has exactly k nonsingular irreducible invariant algebraic curves. Additionally we provide sufficient conditions in order that all the algebraic limit cycles are hyperbolic. We also provide lower bounds for N.

The billiard motion inside an ellipsoid Q Rn+1 is completely integrable. Its
phase space is a symplectic manifold of dimension 2n, which is mostly foliated with Liouville
tori of dimension n. The motion on each Liouville torus becomes just a parallel translation
with some frequency ! that varies with the torus. Besides, any billiard trajectory inside Q is
tangent to n caustics Q 1 ; : : : ;Q n, so the caustic parameters = ( 1; : : : ; n) are integrals
of the billiard map. The frequency map 7! ! is a key tool to understand the structure of
periodic billiard trajectories. In principle, it is well-defined only for nonsingular values of the
caustic parameters.
We present four conjectures, fully supported by numerical experiments. The last one gives
rise to some lower bounds on the periods. These bounds only depend on the type of the
caustics. We describe the geometric meaning, domain, and range of !. The map ! can
be continuously extended to singular values of the caustic parameters, although it becomes
“exponentially sharp” at some of them.
Finally, we study triaxial ellipsoids of R3. We compute numerically the bifurcation curves
in the parameter space on which the Liouville tori with a fixed frequency disappear. We
determine which ellipsoids have more periodic trajectories. We check that the previous lower
bounds on the periods are optimal, by displaying periodic trajectories with periods four, five,
and six whose caustics have the right types. We also give some new insights for ellipses of R2.

For a polynomial planar vector field of degree n ≥ 2 with generic invariant algebraic curves we show that the maximum number of algebraic limit cycles is 1 + (n − 1)(n − 2)/2 when n is even, and (n − 1)(n − 2)/2 when n is odd. Furthermore, these upper bounds are reached.